An 8-manifold of special holonomy Spin(7).
Equivalently: an 8-manifold equipped with a globalization of the Cayley 4-form.
classification of special holonomy manifolds by Berger's theorem:
(Leung 02)
Spin(8)-subgroups and reductions to exceptional geometry
see also: coset space structure on n-spheres
On a spin-manifold of dimension 8 a choice of topological Spin(7)-structure is equivalently a choice of cocycle in J-twisted Cohomotopy cohomology theory. This follows (FSS 19, 3.4) from
the standard coset space-structures on the 7-sphere (see here)
the fact that coset spaces are the homotopy fibers of the maps of the corresponding classifying spaces (see here)
Let be a closed smooth manifold of dimension 8 with Spin structure. If the frame bundle moreover admits G-structure for
then the Euler class , the second Pontryagin class and the cup product-square of the first Pontryagin class (the combination proportional to the I8-term) of the frame bundle/tangent bundle are related by
The same conclusion (1) also holds for Spin(5).Spin(3)-structure, see there.
See also at C-field tadpole cancellation.
The concept goes back to:
Characterization in terms of the Euler 8-class and the I8-invariant polynomial of the tangent bundle:
Construction of compact Spin(7)-manifolds:
Christine Taylor, Compact Manifolds with Holonomy Spin(7) (1996) (pdf)
Dominic Joyce, A new construction of compact 8-manifolds with holonomy , J. Differential Geom. Volume 53, Number 1 (1999), 89-130 (euclid:jdg/1214425448)
Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford Mathematical Monographs, Oxford University Press (2000) (ISBN:9780198506010)
In terms of G-structure:
and motivated from special supersymmetry (such as in M-theory on Spin(7)-manifolds):
Chris Isham, Christopher Pope, Nowhere Vanishing Spinors and Topological Obstructions to the Equivalence of the NSR and GS Superstrings, Class. Quant. Grav. 5 (1988) 257 (spire:251240, doi:10.1088/0264-9381/5/2/006)
Chris Isham, Christopher Pope, Nicholas Warner, Nowhere-vanishing spinors and triality rotations in 8-manifolds, Classical and Quantum Gravity, Volume 5, Number 10, 1988 (cds:185144, doi:10.1088/0264-9381/5/10/009)
(focus on Spin(7)-structure)
Relating M-theory on Spin(7)-manifolds with F-theory on Spin(7)-manifolds via Higgs bundles:
Last revised on April 28, 2024 at 13:52:18. See the history of this page for a list of all contributions to it.